If you multiply a number with only 1s by itself, you end up with a palindrome of the numbers in ascending and then descending order. People have called this an example of the “beauty of mathematics.” If your number has N digits of 1, then its square will be the numbers 1 to N in ascending and then descending order. The pattern continues indefinitely, although it “breaks” after 9 because of carry over. I explain why the pattern happens using the method of multiplying by lines.

Multiply by lines: https://www.youtube.com/watch?v=0SZw8jpfAk0

Blog post (text summary): http://wp.me/p6aMk-4pp

If you like my videos, you can support me at Patreon: http://www.patreon.com/mindyourdecisions

Connect on social media. I update each site when I have a new video or blog post, so you can follow me on whichever method is most convenient for you.

My Blog: http://mindyourdecisions.com/blog/

Twitter: http://twitter.com/preshtalwalkar

Facebook: https://www.facebook.com/pages/Mind-Your-Decisions/168446714965

Google+: https://plus.google.com/108336608566588374147/posts

Pinterest: https://www.pinterest.com/preshtalwalkar/

Tumblr: http://preshtalwalkar.tumblr.com/

Instagram: https://instagram.com/preshtalwalkar/

Patreon: http://www.patreon.com/mindyourdecisions

Newsletter (sent about 2 times a year): http://eepurl.com/KvS0r

My Books

Here’s a listing of all my books

http://goo.gl/BDlEkB

Here’s a more detailed description of each book…

“The Joy of Game Theory” shows how you can use math to out-think your competition. (rated 4/5 stars on 21 reviews) http://amzn.to/1uQvA20

“Math Puzzles Volume 1” features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.6/5 stars on 9 reviews. http://amzn.to/1GhUUSH

“Math Puzzles Volume 2” is a sequel book with more great problems. http://amzn.to/1NKbyCs

“Math Puzzles Volume 3” is the third in the series. http://amzn.to/1NKbGlp

“40 Paradoxes in Logic, Probability, and Game Theory” contains thought-provoking and counter-intuitive results. (rated 4.9/5 stars on 7 reviews) http://amzn.to/1LOCI4U

“The Best Mental Math Tricks” teaches how you can look like a math genius by solving problems in your head http://amzn.to/18maAdo

“Multiply Numbers By Drawing Lines” This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. http://amzn.to/XRm7M4

what if you have more tha 20 ones??

101^0 = 01

101^1 = 0101

101^2 = 010201

101^3 = 01030301

101^4 = 0104060401

101^5 = 010510100501

101^6 = 01061520150601

…

Awesomeeeee!!!

I Can Turn Any Number Into Four Say For Example The Number 20

20 = 6 = 3 = 5 = 4 Because 20 Has 6 Letters In It. Here Count The Letters In Twenty. Repeat This For Every Number It Always Will Equal 4

I've just noticed that your method with the line intersections works not only for squaring but also for multiplying numbers consisting of any count of digit 1. For example 11 * 111 = 1221 , 11 * 1111 = 12221 , 111 * 1111 = 123321 …and so on.

EW COMMON CORE.

Gosh i realised this aswell

i feel nerdz bc i love to watch these videos

Here's another interesting thing. If you know about binomial coefficients, you will instantly be able to calculate 11^4. Why? Because 11^4 = (10+1)^4=14641

Simmilarly 101^8 = 10828567056280801

Which can be calculated by simply looking up in pascal's triangle

if you keep on going as far as the 28th term, the counting pattern actually continues!

123456790 123456790 123456790 098765432 098765432 0987654321

^ ^

all of the sections have all numbers 1-9 except for these. That is because all the sections in the first half have 1-9, and the very last section has 1-9, but all the others don't. This is because we carry over to the left and not the right.

Another way of looking at 111×111:

111×100=11100

111×10= 01110

111×1= 00111

when it is added together it is 12321 (doesn't it look similar to the line thing?)

Here's another way to do it:

1 x 1 =

1

x1

1

And:

111 x 111 =

11100

01110

x 00111

12321

Crap this is intresting. Discovered something simmilar 5 years ago, but forgot that later.

This happens in every base other than decimal too

I always assumed this trick worked because 11 is one more than 10 (the base for our system) but I now see that that is just a coincidence. The reputation of the unit (1) is really what makes this work. Great video!

How about the pattern of the blue numbers in the middle with 10 ones or higher? They go 00, 0120, and is 012340 next?

+MindYourDecisions can you please make a fancy video for the proof of the fundamental theorem of algebra?

Hello mindyourdecisions. i was existed to watch this new video, but i cant watch it. do you know why and how I can fix this? thanks in advance.

It's really amazing. Btw, this lines drawn in 2d, will it work in 3d, with multiply 3 numbers?

I wish we we would switch to base 12. THEN you'd see a lot of fun patterns. And 'decimal' approximations for common ratios would be less necessary (there would still be dodecimal approximations when we have weird things but at least thirds would be 'whole' again in common math)

There's another pattern I have found. When you share

9 / 1

98 / 12

987 / 123

9876 / 1234

…

When you keep sharing this you get closer to 8. Can you explain this?